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Simplifying 16t2 + 29t + 7 = 19 Reorder the terms: 7 + 29t + 16t2 = 19 Solving 7 + 29t + 16t2 = 19 Solving for variable 't'. Reorder the terms: 7 + -19 + 29t + 16t2 = 19 + -19 Combine like terms: 7 + -19 = -12 -12 + 29t + 16t2 = 19 + -19 Combine like terms: 19 + -19 = 0 -12 + 29t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.75 + 1.8125t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.8125t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.8125t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.8125t + t2 = 0 + 0.75 1.8125t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.8125t + t2 = 0.75 The t term is 1.8125t. Take half its coefficient (0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. 1.8125t + 0.8212890625 + t2 = 0.75 + 0.8212890625 Reorder the terms: 0.8212890625 + 1.8125t + t2 = 0.75 + 0.8212890625 Combine like terms: 0.75 + 0.8212890625 = 1.5712890625 0.8212890625 + 1.8125t + t2 = 1.5712890625 Factor a perfect square on the left side: (t + 0.90625)(t + 0.90625) = 1.5712890625 Calculate the square root of the right side: 1.253510695 Break this problem into two subproblems by setting (t + 0.90625) equal to 1.253510695 and -1.253510695.Subproblem 1
t + 0.90625 = 1.253510695 Simplifying t + 0.90625 = 1.253510695 Reorder the terms: 0.90625 + t = 1.253510695 Solving 0.90625 + t = 1.253510695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + t = 1.253510695 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + t = 1.253510695 + -0.90625 t = 1.253510695 + -0.90625 Combine like terms: 1.253510695 + -0.90625 = 0.347260695 t = 0.347260695 Simplifying t = 0.347260695Subproblem 2
t + 0.90625 = -1.253510695 Simplifying t + 0.90625 = -1.253510695 Reorder the terms: 0.90625 + t = -1.253510695 Solving 0.90625 + t = -1.253510695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + t = -1.253510695 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + t = -1.253510695 + -0.90625 t = -1.253510695 + -0.90625 Combine like terms: -1.253510695 + -0.90625 = -2.159760695 t = -2.159760695 Simplifying t = -2.159760695Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.347260695, -2.159760695}
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